The Topology of Probability Distributions on Manifolds

Let $P$ be a set of $n$ random points in $R^d$, generated from a probability measure on a $m$-dimensional manifold $M \subset R^d$. In this paper we study the homology of $U(P,r)$ -- the union of $d$-dimensional balls of radius $r$ around $P$, as $n \to \infty$, and $r \to 0$. In addition we study the critical points of $d_P$ -- the distance function from the set $P$. These two objects are known to be related via Morse theory. We present limit theorems for the Betti numbers of $U(P,r)$, as well as for number of critical points of index $k$ for $d_P$. Depending on how fast $r$ decays to zero as $n$ grows, these two objects exhibit different types of limiting behavior. In one particular case ($n r^m > C \log n$), we show that the Betti numbers of $U(P,r)$ perfectly recover the Betti numbers of the original manifold $M$, a result which is of significant interest in topological manifold learning

Distance two labeling of direct product of paths and cycles

Suppose that $[n]=\left\{0,1,2,...,n\right\}$ is a set of non-negative integers and $h,k \in [n]$. The $L(h,k)$-labeling of graph $G$ is the function $l:V(G)\rightarrow[n]$ such that $\left|l(u)-l(v)\right|\geq h$ if the distance $d(u,v)$ between $u$ and $v$ is one and $\left|l(u)-l(v)\right| \geq k$ if the distance $d(u,v)$ is two. Let $L(V(G))=\left\{l(v): v \in V(G)\right\}$ and let $p$ be the maximum value of $L(V(G)).$ Then $p$ is called $\lambda_h^k-$number of $G$ if $p$ is the least possible member of $[n]$ such that $G$ maintains an $L(h,k)-$labeling. In this paper, we establish $\lambda_1^1-$ numbers of $P _m \times C_n$ graphs for all $m \geq 2$ and $n\geq 3$.Comment: 13 pages, 9 figure

Moments in graphs

Let $G$ be a connected graph with vertex set $V$ and a {\em weight function} $\rho$ that assigns a nonnegative number to each of its vertices. Then, the {\em $\rho$-moment} of $G$ at vertex $u$ is defined to be M_G^{\rho}(u)=\sum_{v\in V} \rho(v)\dist (u,v) , where \dist(\cdot,\cdot) stands for the distance function. Adding up all these numbers, we obtain the {\em $\rho$-moment of $G$}: M_G^{\rho}=\sum_{u\in V}M_G^{\rho}(u)=1/2\sum_{u,v\in V}\dist(u,v)[\rho(u)+\rho(v)]. This parameter generalizes, or it is closely related to, some well-known graph invariants, such as the {\em Wiener index} $W(G)$, when $\rho(u)=1/2$ for every $u\in V$, and the {\em degree distance} $D'(G)$, obtained when $\rho(u)=\delta(u)$, the degree of vertex $u$. In this paper we derive some exact formulas for computing the $\rho$-moment of a graph obtained by a general operation called graft product, which can be seen as a generalization of the hierarchical product, in terms of the corresponding $\rho$-moments of its factors. As a consequence, we provide a method for obtaining nonisomorphic graphs with the same $\rho$-moment for every $\rho$ (and hence with equal mean distance, Wiener index, degree distance, etc.). In the case when the factors are trees and/or cycles, techniques from linear algebra allow us to give formulas for the degree distance of their product

Mathematical model of coordination number of spherical packing

The article considers a mathematical model of the coordination number, which allows obtaining an equation for multi component spherical packing in the entire range of its change. The resulting model can be used in both 2-d and 3-d spaces. The concept of the coordination index is introduced as a function of the inter-particle distance related to a single particle located near the central particle. The model provides unambiguous compliance between the simulated and calculated data on the coordination numbers of the spherical packin

On Multiplicatively Badly Approximable Vectors

Let $\|x\|$ denote the distance from $x\in\mathbb{R}$ to the set of integers $\mathbb{Z}$. The Littlewood Conjecture states that for all pairs of real numbers $(\alpha,\beta)\in\mathbb{R}^{2}$ the product $q\|q\alpha\|\|q\beta\|$ attains values arbitrarily close to $0$ as $q\in\mathbb{N}$ tends to infinity. Badziahin showed that, with an additional factor of $\log q\log\log q$, this statement becomes false. In this paper we prove a generalisation of this result to vectors $\boldsymbol{\alpha}\in\mathbb{R}^{d}$, where the function $\log q\log\log q$ is replaced by the function $(\log q)^{d-1}\log\log q$ for $d\geq 2$, thereby obtaining a new proof in the case $d=2$. As a corollary, we deduce some new bounds for sums of reciprocals of fractional parts.Comment: Appendix C by Reynold Fregoli and Dmitry Kleinboc